Generally, ceramic fibers, and structures containing such fibers, are formed by a sol-gel process ceramic. In such a process, a sol-gel composition is coated and/or deposited onto the surface of, for example, an organic polymer fibers, and then the resulting combination is subjected to oxidation to yield a ceramic fiber. In more detail, a coated fiber is heated in air to cause the sol-gel composition to chemically react, and to oxidize the underlying polymer support, thereby essentially removing the polymer support material from the fiber (except for trace amounts of carbon). However, fibers created by such a process tend to suffer a number of drawbacks. For example, such fibers tend to be brittle, thus limiting their usefulness. Accordingly, there is a need in the art for flexible ceramic fibers, and for a process for making such fibers.